Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486333 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18194 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
933 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 933 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,7)(5,19)(6,11)(8,13)(9,17)(10,14)(15,23)(16,26)(18,21)(20,27)(22,29)(24,30)(25,28)(31,32);(2,7)(3,13)(4,17)(5,14)(8,23)(9,10)(11,21)(12,20)(15,30)(16,28)(18,29)(19,22)(24,25)(26,31);(4,18)(15,30)(17,29)(27,32);(1,3,14,28,6,16,5,13)(2,8,19,26,11,25,10,12)(4,15,29,32,18,30,17,27)(7,20,9,24,21,31,22,23)Polycyclic presentation:
2^9|d^2=gi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,g^b=gi,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231612, #72231701, #72231710, … |
| maximal normal subgroup of | #72231612, #72231701, #72231710, … |
| maximal quotient of | #72250318, C2×#486335, #72265142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393531);